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During the 2000s, over a dozen U.S. states passed laws that prohibit healthcare employers from mandating overtime for nurses. Using a nationwide panel data set from 2004 to 2012, we find that these mandatory overtime laws reduced the service quality of nursing homes, as measured by an increase in deficiency citations. This outcome can be explained by two undesirable changes in the staffing hours of registered nurses: decreased hours of permanent nurses and increased hours of contract nurses per resident day. We observe that the increase in deficiency citations concentrates in the domains of administration and quality of care rather than quality of life, and the severity levels of the increased citations tend to be minor rather than major. We also find that the laws’ negative effect on quality is more severe in nursing homes with higher percentages of Medicare-covered residents. These observations are consistent with the predictions of a stochastic staffing model that incorporates demand uncertainty and operational flexibility. Furthermore, we rule out an alternative hypothesis that the quality decline is induced by an increase in nurse wages. This paper was accepted by Gad Allon, operations management.

In this paper, we consider the design of a synchronous, unpaced flow line where workers operate at different skill levels and overtime is used, if necessary, to meet a daily production quota. The line is unpaced in the sense that items only move to the next workstation when all workers on the line have completed their respective tasks. The design problem in this case is to assign both workers and tasks to workstations to minimize the expected sum of regular and overtime costs. To solve this problem, we develop an optimization algorithm for smaller problems and a heuristic algorithm for larger problems, which we use to investigate the sensitivity of total expected cost to changes in the price of overtime, hiring practices, worker differences, and the overall amount of work time variability. Based on an extensive computational analysis, we found that (1) planned overtime is frequently beneficial, (2) more workers should be hired as worker variability increases, and (3) increases in overtime costs frequently yield a relatively lower percentage increase in total expected cost. Other managerial implications are discussed in the paper.

We consider the situation of a supplier plant whose customer plants desire just-in-time (JIT) deliveries. Randomness in both the production and demand processes make satisfying every demand that might occur in true JIT fashion impossible. Therefore, supplier plants typically negotiate bounds on JIT contracts with their customers. In this paper, we focus on the use of “quotas” or “target inventory levels” as a mechanism for establishing such bounds. That is, the supplier firm is responsible for meeting periodic demands up to the quota, but not beyond. In this paper, we consider the problem of setting an appropriate quota from the perspective of the supplier plant and interpret our results in the context of the negotiation process between the supplier and its customers. Under the assumption that “safety capacity” (i.e., overtime or a vendoring option) is available, we develop two models that address this problem. The first model assumes that quota shortfalls cannot be carried over to the next regular time production period and are made up with overtime/vendoring, which incurs fixed plus variable costs. The second model assumes that shortages can be backlogged to the next regular time production period at a cost. We use numerical examples to demonstrate how the models we developed were used by a clutch supplier to a large auto manufacturer to negotiate its JIT contracts.

This paper presents an integer programming model for developing optimal shift schedules while allowing extensive flexibility in terms of alternative shift starting times, shift lengths, and break placement. The model combines the work of Moondra (Moondra, S. L. 1976. An L. P. model for work force scheduling for banks. J. Bank Res. 7(4) 299–301.) and Bechtold and Jacobs (Bechtold, S. E., L. W. Jacobs. 1990. Implicit optimal modeling of flexible break assignments in labor staffing decisions for service operations. Management Sci. 36(11) 1339–1351.) by implicitly matching meal breaks to implicitly represented shifts. Moreover, the new model extends the work of these authors to enable the scheduling of overtime and the scheduling of rest breaks. We compare the new model to Bechtold and Jacobs' model over a diverse set of 588 test problems. The new model generates optimal solutions more rapidly, solves problems with more shift alternatives, and does not generate schedules violating the operative restrictions on break timing.

Growing interest in designing systems with capacity slack as one form of flexibility raises many questions about its use and its usefulness. In the framework of the economic lot scheduling problem with stochastic demand, we develop on optimization-based model that considers capacity slack, safety stock, and overtime explicitly, and has the objective of minimizing the expected cost per unit time of inventory, overtime, and, where applicable, setup costs. The solution is a continuous-time production plan that consists of a time-dependent inventory trajectory for each of the parts, including the placement of planned idle time in the schedule. We consider schedule stability to be desirable because of potential effects on upstream and downstream operations in multistage production settings. Thus, the plan also has certain characteristics that contribute to achieving stability. Our results on the relative merits of capacity slack and safety stock indicate that capacity slack in the form of planned idle time is not a cost-effective hedge against demand uncertainty in this context. Thus, it is essential that management carefully identify and evaluate other reasons for including idle time in a plan, and use the idle time effectively. For managers who face situations not fully represented by our model, this paper provides analytic results that will guide them in the placement of planned idle time and the choice of safety stock levels.

This paper presents simple approximations for networks of queues with overtime operation at some stations. This type of network is commonly encountered in several manufacturing applications. We provide bounds on the performance of the approximations for single and multiple machine stations. Our results suggest that the methods perform satisfactorily. These approximations can be used in conjunction with parametric decomposition methods to analyze queueing networks. The computational results indicate that the performance of the decomposition approach does not deteriorate when combined with the methods proposed in this paper.

For purposes of aggregate production smoothing, quadratic functions often produce good fits to data on some of the relevant incremental costs, such as inventory and overtime costs. However, when costs of hiring and laying off workers are described by smooth quadratic functions, the model solutions tend to involve very frequent minor workforce adjustments. Production managers are often unwilling to actually implement such decisions because changing workforce levels involves a great many intangible costs, many of which seem to defy meaningful analytic representation. In this paper an interactive model is developed where, through parametric analysis of a set of workforce constraints, a manager can establish the appropriate trade-offs between his judgmental estimates of hiring and firing costs and other more tangible production costs.

Setup reduction is an important aspect of the Just-In-Time (JIT) and Zero Inventory (ZI) concepts. This paper looks at the increased effective capacity that results from setup reduction. A certain amount of time, as well as expense, is required for each setup within a multiproduct firm. By reducing the time required for setups, less time is required to carry out those setups. That time can either be used to reduce lot sizes or to reduce overtime. We use an EOQ-like setting with stationary deterministic demand rates for each product. This simple model allows explicit solutions to be derived, not only for optimal setup time reduction, but for optimal overtime as well. Numerical examples illustrate the results.

In this paper we consider a certain aggregate production planning model. This model permits regular and overtime production and allows for backordering of goods for a number of periods. Although the discussed model can be formulated as a linear programming problem a special (noniterative) method is developed. First the optimal level of each mode of production is determined for each period. Then the complete solution is immediately constructed. This procedure is so simple that it can be also implemented via pencil and paper. Several computational improvements as well as problem variations are discussed.

In Part II, the CLSP Heuristic of Part I is extended to include variable capacity constraints—in this case the ability to adjust workforce levels and to employ overtime. The paper concludes with computational results for both the fixed and variable capacity heuristics.

This paper examines issues in building decision support models for budgeting nursing workforce requirements in a hospital. We determine regular-time, overtime, and agency workforce levels for various skill classes in a budget cycle. We introduce a family of eight models ranging from a single-period, aggregate and deterministic model to a multiperiod, disaggregate and probabilistic model. In a single-period model, we ignore the time-varying nature of demand for nursing hours. Aggregation is done over the nurse skill class mix. For probabilistic models, we consider demand uncertainty. Using empirical data, we evaluate the effects of level of sophistication in model building and in information requirements on their relative performances. The results suggest that ignoring the time-varying nature of demand does not induce gross errors in budget estimates. However, ignoring demand uncertainty produces underestimates (about five to six percent) of budget needs—a consequence of a Madansky (Madansky, A. 1960. Inequalities for stochastic linear programming problem. Management Sci. 6 197–204.) inequality. It also induces added costs to the system due to implementing nonoptimal regular-time workforce levels. Finally, we find that a simple formula using a single-period demand estimate gives excellent approximations to the budget estimates obtainable from the more precise models.

A scheduling procedure is proposed for the static job shop problem with job due dates and overtime capability. An example is used to demonstrate the procedure and the generation of trade-off curves relating overtime and total tardiness for use in managerial evaluation of alternative schedules. Results are reported for a set of trial problems having 30 to 98 processing operations. An extension of the procedure is described for dynamic job shop scheduling.

When a demand pattern is dominated by a seasonal effect, the concept of a steady state solution can be used in two ways in aggregate production planning. First, general policy recommendations can be made concerning the use of seasonal workforce changes versus overtime and seasonal inventory. Second, the results can be used to provide ending conditions in an intermediate range planning algorithm with a moving horizon. These ideas are explored using a linear programming model, making use of known planning horizon properties. A simulation experiment tests the efficacy of short horizon deterministic models in a stochastic environment, and demonstrates that ending conditions, derived from the steady state model, improve decisions under a variety of conditions on costs and horizon length.

In Part I of this paper, we developed an algorithm for finding planning horizons for the deterministic production smoothing problem when all demand was required to be met from regular production, under rather general assumptions for the production, production smoothing, and holding cost functions. Here, in Part II, we develop analogous planning horizons when the model is extended to include such factors as overtime, lost sales, simple subcontracting, undertime, and backlogging. Part II depends heavily on Part I. To avoid unnecessary tedium, all cost functions will be assumed to be stationary in Part II. Overtime, undertime, holding, and backlogging will be assumed to be convex increasing, all other costs linear and proportional. However, it is not difficult to see that the techniques of Part I can be utilized directly to cover more general cost functions. Part II concludes with a discussion of the application of our general approach to actual data.

We calculate the expected additional fire protection cost to a city implied by the 1974 amendments to the Fair Labor Standards Act. Under the amendments, if a city chooses not to pay its men who are on duty for more than 24 hours at a time for sleep and meal time, it may exclude up to eight hours of sleep time, but must compensate them at overtime rates for the entire sleep period if one or more alarms prevent them from obtaining at least five uninterrupted hours of sleep in that period. Interalarm times are modeled as a Poisson process, and both point and interval estimates of the additional annual overtime cost are computed for the city of Monterey by estimating the annual number of interrupted sleep periods. A model that permits the rate at which alarms are received and the length of die designated sleep period to be varied is developed for computing the proportion of nights in the year during which the firemen get five uninterrupted hours of sleep. A procedure for adjusting the model to allow for the time spent in response to an alarm is also described.

This research focuses on the effect of setup time on lot sizing. The setting is the Capacitated Lot Sizing Problem (the single-machine lot sizing problem) with nonstationary costs, demands, and setup times. A Lagrangian relaxation of the capacity constraints of CLSP allows it to be decomposed into a set of uncapacitated single product lot sizing problems. The Lagrangian dual costs are updated by subgradient optimization, and the single-item problems are solved by dynamic programming. A heuristic smoothing procedure constructs feasible solutions (production plans) which do not require overtime. The algorithm solves problems with setup time or setup cost. Problems with extremely tightly binding capacity constraints were much more difficult to solve than anticipated. Solutions without overtime could not always be found for them. The most significant results are that (1) the tightness of the capacity constraint is a good indicator of problem difficulty for problems with setup time; and (2) the algorithm solves larger problems better than smaller problems, although they are more time consuming to solve. This indicates that larger problems may be easier despite the greater computational effort they require.

This paper studies the planning problem faced by a machine shop required to produce many different items so as to meet a rigid delivery schedule, remain within capacity limitations, and at the same time minimize the use of premium-cost overtime labor. It differs from alternative approaches to this well-known problem by allowing for setup cost indivisibilities. As an approximation, the following linear programming model is suggested: Let an activity be defined as a sequence of the inputs required to satisfy the delivery requirements for a single item over time. The input coefficients for each such activity may then be constructed so as to allow for all setup costs incurred when the activity is operated at the level of unity or at zero. It is then shown that in any solution to this problem, all activity levels will turn out to be either unity or zero, except for those related to a group of items which, in number, must be equal to or less than the original number of capacity constraints. This result means that the linear programming solution should provide a good approximation whenever the number of items being manufactured is large in comparison with the number of capacity constraints.

The problem of production and employment scheduling may be stated as follows. Given the monthly demands for the product turned out by a factory, what should be the monthly production rates and work force levels in order to minimize the total cost of regular payroll and overtime, hiring and layoffs, inventory and shortages incurred during a given planning interval of several months? This problem has received a classical solution in two papers by Holt, Modigliani, Muth, and Simon (Holt, C. C., F. Modigliani, H. A. Simon. 1955. A linear decision rule for production and employment scheduling. Management Sci. (October); Holt, C. C., F. Modigliani, J. F. Muth. 1956. Derivation of a linear decision rule for production and employment. Management Sci. (January).). These authors assumed quadratic cost functions. Their treatment of the problem will be referred to as “quadratic programming.” It appears, however, that in the majority of practical applications and theoretical models the cost functions are assumed to be linear. It, therefore, seems desirable to have a method of solution for the linear case as well. In this paper it is shown that a solution can be obtained by linear programming methods. From the linear programming viewpoint, this paper is of an expository nature. Management Technology, ISSN 0542-4917, was published as a separate journal from 1960 to 1964. In 1965 it was merged into Management Science.

This paper is concerned with the modeling and analysis of a one-machine two-product, integrated production-inventory system. A semi-Markovian model is developed which incorporates a state dependent control system heretofore not used in integrated production-inventory models. Control optimization is achieved through the use of Markov-renewal programming. The model is exercised over a range of parameters in order to determine the characteristics of the system. An analysis of the experimental results leads to the following conjectures about the production-inventory system: 1. For the production-inventory system, the economic manufacturing quantity is greater than that seen in the deterministically derived EMQ. The increase results primarily from increases in expected overtime costs and expected setup costs rather than increases in stockout costs. This is important for applications, since overtime and setup costs are a good deal easier to quantify than stockout costs in a real system. 2. Under some conditions, deterministically derived rules for the application of overtime have merit in a stochastic system.

In this paper we study a model that minimizes the sum of production, employment smoothing, and inventory costs subject to a schedule of known demand requirements over a finite time horizon. The three instrumental variables are work force producing at regular-time, work force producing on overtime, and the total work force. Overtime is limited to be not more than a fixed multiple of regular time. The idle portion of the work force and the levels of inventory are resultant variables. We postulate the following shape characteristics for the cost functions production costs are convex-like, smoothing costs are V-shaped, and holding costs are increasing, both the production and holding cost functions need not be stationary. In this paper, we provide upper and lower bounds on the cumulative regular-time plus overtime work force for any sequence of demand requirements. We also give the form of an optimal policy when demands are monotone (either increasing or decreasing). Finally, we derive the asymptotic behavior of optimal policies when demands are monotone and the planning horizon becomes arbitrarily long. All of these results, which convey information about the numerical values of optimal policies, given specific demands and an initial level of inventory, depend only on the shape characteristics of the cost functions. Algorithmic techniques are discussed elsewhere [Lippman, S. A., A. J. Rolfe, H. M. Wagner, J. S. C. Yuan. Algorithms for optimal production scheduling and employment smoothing. Opns Res. To appear.], [Yuan, J. S. C. 1967. Algorithms and multi-product model in production scheduling and employment smoothing. Technical Report 22 (NSF GS-552), Stanford University, August.].

This paper describes a production smoothing system which combines several known scheduling tools into an operational unit. The input to the system is forecasted customer demand; the outputs are the required production levels, size of labor force, planned overtime and expected inventories of classes and individual end products. Because of varying demand, economic manufacturing quantities are calculated by the method of dynamic programming following Wagner and Whitin. The manufacturing progress function is used to convert units into labor requirements. Finally, the operating schedule is derived by a linear programming formulation which balances payroll costs, the costs of labor fluctuations, and inventory charges. This system has been in operation for several months.

In the case where a given demand increase over a finite time span is known and exceeds current production capacities, a marginal cost analysis can provide decision criteria for work-force adjustments. Using the marginal costs of hiring new workers and for various periods of overtime as a basis, decisions can be made to hire new workers, to work overtime, or to do both. The marginal cost analysis for hiring new workers includes the application of learning curve data to their increasing productivity rate over the time span. The approach minimizes costs, assuming it is not necessary to buy new equipment or add supervisors. The greatest applicability for the model may be found in certain job shop situations.

The scheduled allocation of resources for large-scale harvesting operations is vulnerable to delays caused by variables of a stochastic nature such as rainfall and equipment breakdown. A computer simulation model was developed to test the merits of different harvesting strategies and decision rules on a sugar cane plantation. The model takes into account weather, hauling, equipment logistics, and equipment breakdowns. For a given man-machine harvest strategy and harvest schedule, it will simulate assigning harvest units (men and equipment) to fields, harvesting the cane, hauling the cane to mills, and milling. Submodels, based on a statistical analysis of historical data, generate values of simulated daily rain in each field, equipment breakdown, and repair times. Harvesting delays occur when the simulated rain reaches critical values. The hauling and milling capacity in the model depends to a great extent on both current and previous rainfall and is updated when required for new field conditions brought about by rain. The model keeps account of each cane-hauler round trip (some 50,000 trips a year) from field to mill, causes the haulers to queue when the loader or unloader is busy, and sends them to a repair shop when damaged. Simulated work hours and labor practices are also included in the model. For example, a submodel calls for overtime when progress indicates that the harvest may extend into the rainy season. The model gives periodic reports on the simulated production and information on capacity requirements for many of the tasks within the harvest operation.

Formal manpower planning has received increasing attention in recent years. This paper considers the problem of establishing a manpower policy for defined skill categories in the face of uncertain demands. At one extreme an organization might hire and fire personnel as the demand for their services varies over time. Alternatively, the work force might be held constant at a predetermined level and supplemented with overtime or subcontracting when manpower demand exceeds the supply. Models representing these two strategies are developed and compared. Examples are presented to illustrate their economic implications, and several simple rules of thumb are derived to guide manpower policy decisions.